OPTIMAL DESIGNS FOR RATIONAL MODELS

In this paper, experimental designs for a rational model, Y = P(x)/Q(x ) + epsilon, are investigated, where P(x) = theta(0) + theta(1) + ... + theta(p)x(p) and Q(x) = 1 + theta(p+1) x + ... + theta(p+q)x(q) are polynomials and epsilon is a random error. Two approaches, Bayesian D- optimal and Bayesian optimal design for extrapolation, are examined. T he first criterion maximizes the expected increase in Shannon informat ion provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator (MLE) of the mean response at an extrapolation point x(e). Corresponding locall y optimal designs are also discussed. Conditions are derived under whi ch a p + q + 1-point design is a locally D-optimal design. The Bayesia n D-optimal design is shown to be independent of the parameters in P(x ) and to he equally weighted at each support point if the number of su pport points is the same as the number of parameters in the model. The existence and uniqueness of the locally optimal design for extrapolat ion are proven. The number of support points for the locally optimal d esign for extrapolation is exactly p + q + 1. These p + q + 1 design p oints are proved to be independent of the extrapolation point x(e) and the parameters in P(x). The corresponding weights are also independen t of the parameters in P(x), but depend on x(e) and are not equal.